Developing laminar flow in curved rectangular channels

Developing laminar flow in curved rectangular channels

H.J. de Vriend

Internal report no. 6-78

Laboratory of Fluid Mechanics

Department of Civil Engineering

Developing larninarflow in curved rectangular channels

H.J. de Vriend

Internal report no 6-78

Laboratory of Fluid Mechanics Department of Civil Engineering Delft University of Technology Delft, The Netherlands

.t'

.

,

CONTENTS List of figures List of symbols Sumrnary 1. 1.1. 1.2. 2. 2.I. 2.2. 2.3.

2.4.

3. 3.1. 3.2. 3.3.

4.

4. I.

4.2.

page Introduction . General , .

The present investigation s ••••..•..•...•....•. 2

Mathematical formulation of the problem ...•... 3 Ohannel configuration and coordinate system •.• 3 Differential equation s ..•..•..••..•.•.•...•. 3 Boundary condition s ..•..••••••..•..•...•..•... 4

Normalization ~ 5

Simplification 8

Main and secondary flow; equation of continuit y 8 Simplificatiun of the momentumequations ... 10 'I'rares+ormc.rion to streamwise coordinates ...•.• 12

Main flow computation ..•..•••.•...••.••... 14 Vertical distribution of the main velocity ..•. 14 Depth-averaged main velo ~ity field:' differential

equations 15

4.3. Depth-averaged m~in velocity field: boundary

condi tions 17

4.4. Secondary flow: s tream function equation .•....• 19 4.5. Further simplification of the stream function

equation 22

4.6. Vertical distribution of the stream function of

the secondary flow 23

4.7. Depth-averaged stream function of the secondary

.

flow 25

.

,

page 5. Secondary flow and bed shear stress computation 28 5.1. Extensive computation of the secondary flow •.• 28 5.2. Bed shear stress... 31

6. Verification of the model

...

32 6.1. Verification of the simplifying assumptions

...

32 6.1.1. The equation of continuity for the main flow

..

32 6.1.2. The similarity hypothesis'for the main flow 33 6.1.3. The simplification of the secondary flow

equation 34

6.2. Qualitative comparison wi th turbulent flow

experiments 36

7. Summary of the mathematical model •...•••.. 40

·

,

• J

LIST OF FIGURES

J. Combined cylindrical coordinate systems 2. Definition sketch

3. Streamwise coordinates

4. Applicability of the simplified equation of continuity for the main flow

5. Influence of the streamwise accelerations of the main flow

6.

Effect of the simplification of the secondary flow equation on the main flow

7. Effect of the simplification of the secondary flow equation on the bed shear stress

8.

Effect of the simplification of the secondary flow equation on the secondary flow

·

.

LIST OF SYt-mOLS B C d De Rele: channel width Chézy's factor depth of flow Dean number

Dean number based on the turbulence viscosity

vertical distribution function of the main velocity

esti.mate of

f

g acceleration due to gravity

g vertical distribution function of the stream function ~

e' =

g

*

?

n iteration index

n

horizontal coordinate normal to the streamlines of the

p p p q Q r R r

n

r s R c Vd Re =

v

ReT s

e

S TR T$ u u

oepth-averaged flow pressure

normalized total pressure

scale-factor of the total pressure source term in the equation for g

discharge

norm~lized radial coordinate ra0~al coordinate

norm~lized radius of curvature of the normal lines of the

depth-averaged flow fieLd

normalized radius of curvature of the streamlines of the

depth-averaged flow field

radius of curvature of the channel ax~s Reynolds number

Reynolds number based on the turbulence viscosity longitudinal coordinate

normalized longitudinal coordinate

scale factor of the longitudinal coordinate radial component of the bed shear stress

loneitudinal component of the bed shear stress normalizep longitudinal velocity component depth-averaged value of u

u 'Vm u m u s v V = Q/Bd v v m vR v s v z V<l> 'V v m 'V v s W w m w s y z Cl r e:

=

diR c Z;;

n

v ~ p 0 s/e:Re 'b

,

r '<I> <I> ~ Ijl

-Ijl Ijl.'

₌

Ijl *- Re

normalized longitudinal component of the ma1U velocity normalized streamwise component of the main velocity normalized longitudinal component of the secondary flow normalized radial velocity component

velocity scale

depth-averaged value of v

normalized radial component of the ma1n velocity radial velocity component

normalized radial component of the secondary flow vertical velocity component

longitudinal (tangentia~) velocity component

normalized transverse component of the main velocity normalized transverse component of the secondary flO\-1 normalized vertical velocity component

normalized vertical component ·of the ma1n velocity normalized vertical component of the secondary flmv distance to the left bank

vertical coordinate

deviation angle of the bed shear stress vector with respect to the channel aX1S

curvature ratio

rcrmalized vertical coordinate dynamic viscosity of the fluid kinematic viscosity of the fluid normalized radial coordinate mass density'of the fluid

longitudinal coordinate

normalized total bed shear stress

normalized radial component of the bed shear stress

normalized longitudinal component of the bed shear stress

angular coordinate

stream function of the depth-averaged ma1n flow stream function of the secondary flow

I'

"

w vorticity of the depth-averaged main fLow

'"

vorticity of the main flow

w ,w m m

w vorticity of the secondary flow s

• J

SU~~y

As an intermediate step between earlier investigations on fully developed laminar flow in curved channels of shallow rectaneular wet cross-section and the mathematical modeling of turbulent flow in river bends, a mathematical model of developing laminar flow in such channels is investigated. The most important assumptions made in addition to those based on the fully developed flow investigations are

verified for the flow ~n a rather sh~rply curved flume with rather strong effects of curvature. Experimental data on turbulent flow are compared qualitatively with the results of an equivalent laminar flow computation for two different flumes. This comparison shows the measured and the computed depth-averaged main velocity distributions to agree weIl beyond the first part of the belld, which confirms the presumption that the mechanism of the main velocity redistribution in turbulent flow is essentially the same as in laminar flow.

I

-.

,

Iiltroduction

l.I. General

Introduction

The flow and the bed topography in curved alluvial river channels play a prominent part,in several aspects of river engineeripg, such as navigability, bank protection and dis-pers ion of pollutants. Hitherto, engineering problems con-cerning r1ver bends are mostly investigated using physical scale modeIs, even though the complex character of the flow may g1ve rise to scale effects making the model data hard to interpret in prototype terms. The increased facilities of electronic computers, however, make mathematical models attractive to be developed. They would facilitate the understanding of the physical phenomena and could be used

together with or even instead of physical roodels.

As it is impossible to reliably predict the bed topography without knowing the flow field, an adequate model of the flow in a curved channel with an uneven bed topography must be developed first Assuming disturbances of the flow to

travel at a much higher celerity than disturbances of the bed,. as is the case in most of the navigable alLuvxaI rivers, the bed can be considered as being fixed when computing the flow. In addition, the flow can be assumed to be steady, which is allowable under many practical condi.t i.ons.

The development of a mathematical model of steady flow 1n river bends with a fixed uneven bed forms one of the research projects of the Laboratory of Fluid Mechanics of the Delft University of Technology, as a part of the river

~

bend project of the joint hydraulic research programme T.O.W. )

*) ,_{'Toegepast Onderzoek Waterstaat", in which Rijkswaterstaat,}

the Delft Hydraulics Laboratory and the Delft University of Technology collaborate.

• J 2

-1.2. The present investigations

The present investigations, concerning a mathematical

model of developing laminar flow in curved shallow channels, form an intermediate step between the investigation of fully developed laminar flow in such channels (DE VRIEND, 1978) and the mathematical.modeling of turbulent flow in shallow river bends.

This intermediate step was made in order to develop û

computational procedure for the solutión of the mathematical

system describing this complex type of flow. Although most

important information on the possible outlines of such a

procedure was obtained from the fully developed (axisymmetric)

laminar flow investigations (DE VRIEND, 1978), further research

is needed as regards the phenomena that are cssentially

non-axisymmetric. In addition, the extra (streamwise) dimension

makes the mathematical system more complicated and it must be

investigated to what extent the computational techniques utilized

in the fully developed flow model are still applicable.

The laminar flow typ~ was chosen because it provides a logical

continuation of the work on fully developed laminar flow.

Further-more, the arguments that led to the Lnvesti.gati.v-n of fully

developed laminar flow hold here, as weIl: its mathematical

description is simp Ier and contains less uncertainties than a

turbulent flow model, whereas the mechanisms of the most important

curvature effects (secondary flow; main velocity redistribution)

are essentially the same in laminar and turbulent flow (see also:

.1' ₃ -• J

2. Mathematica1 formu1ation of the prob1em

2.J. Channe1 configuration and coordinate system

The mathematica1 model to be deve10ped for the prediction of flow in curved river channe1s shou1d account for a free surf ace and more or 1ess arbitrary channe1 patterns and bed configurations. In the

present part of the investigations, however, on1y the essentia1 features of a computation method for deve10ping curved flow are concerned. There-fore considerations are 1imited to channe1s of uniform rectangu1ar wet

.

cross-section, with a channe1 axis consisting of circu1ar arcg with radii

x

of curvature R ). According1y, the coordinate system to be used for

. ck. .

the present computat10ns cons1sts of~a set of cy1indrica1 coo~dinate systems, each of which has a vertica1 axis that goes through the centre of the circ1e forming the channe1 axis of the relevant section

(see figure J). On the other hand, a system of curvi1inear, stream-oriented coordinates wi11 be preferab1e in case of an arbitrary channe1 pattern. For the sake of simp1icity,the explanation of the model and the underlying assumptions wi11 be 1imited to one channe1 section with a circular axis of radius R , using a cylindrica1 coordinate system (R,

cp,

z) with vertical

c

z-axis and z

₌

0 at tl-e surface (see figure 2). Only if necessary, a

transformati0h to curvilinear, streamwise coordinates will be carried out.

2.2. Differential equations

The basis of the mathematical model is formed by the conservation laws for mass and momentum for steady laminar flow of an incompressible fluid. If g denotes the acce1eration due to gravity, p the mass density of the f1uid,

V

its kinematic viscosity, p the pressur_ and vR'

Vcp

and

V

z the velocity components in R-,

cp

.

-

and z-direction, respectively, these conservation 1aws yield

o

(2. J)

x) R may be chosen infinitely large, so that the relevant k-th channe I ck -.

,.

₄

-••

(2.2)

(2.3)

v,/,éN'z êv êv 2 --~ --- + v z + v z - -

!

~-

g + v.V v R êcp R êR z êz - p dZ ., Z

(2.4)

2.3.

Boundary cop.ditions

The boundary eondition arisingfromthe impermeability of the surfaee

reads

o

(2.5) .

The vanishing of the shear stress along the surfaee leads to

~vzcpl.=0; ,êvR'1 =0 f} z=O êz 'z=O

(2.6)

The usual dynamie eondition for the pressure at a free surfaee, p

=

0,

ean not be applied here, as the surfaee, whieh must be eonsidered as a

frietionless rigid plate, exerts normal stresses on the fluid.

At

the fixed boundaries the boundary eonditions stemming from the

impermeability and the no-slip eonditions are

vR =

q;

'

vcp= 0; v

=

0

• J 5

-In addition to these "lateral" boundary conditions, inflow and outflow conditions must be given. Most of these conditions will

he formulated in a later stage, only the discharge

Q

is mentioned

here, since it plays a part in the integral condition of continuity

(cf. equation 2.1)

R +B/2

:

c f

R -B/2

·'.c

o

dR f' v dz -d <I>

Q

(2.8)

2.4.

Normalization

The normalization carried out rn order to makeiall. variables

dimensionless and to estimate the order of magnitude of the various

terms in the differential equations is for the greater part the same

as the one applied to the axisynnnetric laminar flow case (DE VRIEND,

1978). In summary: Vu V ~ V' v R ' z c d V-w R c 'with V ""

.9_

_

:Bd

(2.9)

1 R R- Rc

a

I

a

di';, so - = - • aR cl,~'

a

z

=

dz; , so

az

d1

a

al;; (2.10)

It should be noted here that, when the ma~n velocity is defined such

that it is directed along the streamlines of the depth-averaged floT'

field, vR may include not only the radial component of the secondary flow,

but also the radial componen~ of the main velocity. Both components,

however, will he induced hy curvature, so that

(2.9)

is an appropriate

normalization for develo ing as well as ior fully developed curved flow.

The normalization of the pressure and the tangential coordinate requires

further attention. Suppose the total pressure is normalized hy

p + pgz

Pp

(2.11 )

and the tangential coordinate hy

s

=

R(<I>- <1>0)=Ss, whence• 1

a

R

él<l>

1

a

- 6

-• J

,,then the scale-factors Pand S remain to he chosen. Considering

the normalized equation of continuity

v

au

v

+-S as R c (av + ~ ~) +

v

aw a~ R r R aÇ c c

o

_(2.13)

, the first term is likely to he of the same order of magnitude as

V

V

the other ones. Therefore the scale-factors

S

and Rare chosen

c equal, so that S

=

R c 1 all a; whence - -

=

----R a4> R r a4> c (2.14)

The system of equations should include the limit case of fully

developed straight channel flow, for which the longitudinal momentum

equation (2.2) reduces to 2 2

a

V4>

a

v<j> \)(-- + --) az2 ay2

o

- --1 ap + p as (2.15)

As in this equation !he pressure gradient term and the viscous diffusion

terms are of the same importance, the scalefactors for these two types

of terms are choten equal, so that

R

2

v

c

pS

=

pV Vd

cl

(2.16)

Defining E

=

diR and Re

=

Vd/\), the norm~lized system of differential

c

equations ,and houndary conditions then reads

1 au + av + ~ v + aw

=

0

r a4> a~ r aÇ (2. 17)

R .(u au + v au +w ~ + c- uv)

=

E..•e.

r ~

a ~

az;;

<- r

I' • J 7 -.3R E: e (2.19)

3

u aw

aw

aw

~

Re (~--

+

v --

+ w --)

r a~

a~

aÇ

(2.20)

B/2d

"

0

B

f

d~

f

udr;

::::;-,...:;,_

-B/2d

-1 d

0;

au

0;

av

0

0

w

= -- = -= at ç

ar;

a

r;

(2.21) (2.

zz)

u

=

0; v

0: w

=

0

at

r;

-1 and at ~ +!_ - 2d (2.23)

8

-..

3. 'Simplification

3.1. Main and secondary flow; equation of continuity

The normalized system (2.17) through (2.23) will be simplified on the basis of the conclusions drawn from the computations of fully developed laminar flow in curved shallow channels (DE VRIEND, 1978). To that end, main and secondary velocity components will be defined in developing flow and the conclusions referring to the tangential and the transverse velocity components in the fully developed fLow case are assumed to apply to these main and se,condary velocity components,

respectively.

The main flow ~s defineu such, that its horizontal component is the component of the velocity in the direction of the streamlines of thc depth-averaged flow (see figure 3); the secondary flow is defined in planes normal to these streamlines. In the limit case of fully developed flow, the ma~n velocity defined in this way becomes tangential and the secondary flow occurs in the transverse plane, which is consistenti:with the above assumption.

According to this definition, the vertical distribution of the main flow may vary al ng a streamline, which implies that the main flow may have a non-zero vertica: component. Therefore the following separation between the main apd thç secondary flow is made in the mathematical model:

2

u

=

u + EU, V

m s Vm +v,s w=w m +w s (3.1) *)

in which u ,v and w are'defined such, that mmm

ow

m + _.. -ol;;

o

and v m u m v _(3.2) u

~ and ~ denoting the depth-averaged values of u and v, respectively. Consequently, the equation of continuity for the secondary flow reads

2

oU

oV

oW

E S s+~v + __ s=O

r.- ~

+ ~ r s al;;

(3.3)

*)

The tangential component of the secondary flow must be of the

. 2

order O(E ), since both the secondary flow and the deviation of the main flow direction from the tangpntial direction are of the order

·1'

9

-.

,

Thus two separate equations of continuity are obtained,.one for the main flow and one for the secondary flow.

In combination with the depth-averaged equation of continuity

au + av + ~ ~

=

0 r a<j> aE; r (3.4) equations (3.2) lead to au av u u u v 1 m m _{+~v ~!_ (_!!l_) !_ (_!!l_)} 0; m m --- + v =

r a<j> aE; r m r ê<j> - êE;

-u u u v

(3.5)

According to the lat ter equation, the two horizontal components of the main velocity have the same vertical di3tribution; in the former equation the last two terms represent the streamwise variation of this verti::al distribution. Assuming this variation to be negligible with respect to the other terms in this equation, the equation of

continui ty for tlie main flow becomes

o

(3.6)

So neglecting the streamwise variation of the vertical distrioution of the ma1n flow is equivalent to neglecting the vertical component of the ma1n velocity.

One of the possible sirnpLi.fi.cati on. arising from the fully developed

flow computations (DE VRIEND, 1978) is the similarity approximation of the horizental main velocity components. When transposed to developing flow, this reads

u

m u f(r.; s) and vm v f(r.; s) (3.7)

in which

s

denotes the streamwise coordinate. Then the approximation of the first equation Qf

(3

.

5)

by equation

(3.6)

implies that

f

depends only weakly on

s,

i.e. the streamwise derivative of the main velocity 1S

10

-..

3.2.

Simplification of the momentum equations

Substituting (3.)) with ,.,

=

0 into the normalized momentum equations m

(2.18)

through

(2.20)

and neglecting terms being an order O(E

2)

smaller

than the leading terrns of the same type (i.e. advection terms or Qiffu8ion term~) in the relevant equation, the equations reduce to

u oU au ou oU ERe (...1!! _2! + _v m E _{+ V} m m _{+~v u )} o~ +- V U -- + w

äÇ

= r o<p . m r mrn s o~ s r s m 1 op _r;/ (3.8) -- + u r 0<1> 1 m 3 _{(...1!! ~}u ov _{+ -~+}ov ovm ovm um ov8 ov;s E: Rè V V -- + W +--- + V ~ + r o<p m o~ s o~ s ol; r 0<p m + v S

o~

ov s

(3.9)

u ow oW oW 3w E3Re (~_~s + v __ s + v __ s + w s)

r

0<1> m

o~

s

a~

s

aç-.i, (3.

10)

1n which

In fully developed curved shallow channel flow the transverse inertia of the secondary flow, i.e.i.the terms with

ov ow av ow

8 8 S

and s

v:

élE; v o~ w al; w ol;

8 8 8 S

in equations

(3.9)

and

(3.10),

appears to be negligible (DE VRIEND,

1978).

It 18 plausible to neglect these terms in developing flow, as weIl.

As to the advective influence of the ma1n flow on the secondary flow, represented by the terms containing

u

OV

m s

r~

U aw ow ~ __ s and v s r o<p m ~

·

,

- I I

-fully develpped flow case, these terms being identically equal to zero then. A global indication of t~e importance of these terms 1n developing laminar flow can be obtained by considering the decay of a given secondary circulation in a straight reach downstream of a bend in a shallow channel (see Appendix I). In that case the length over which the secondary flow is reduced to 10% of its magnitude at the exit of the bend can be expressed by

10•1 ~ 0.14 Re d (3.11)

So for the Reynolds numbers to be considered here (up to a few hundreds; cf. DE VRIEND, 1978), this length will range up to some

10-20 times the depth of flow. This implies tha~ for shallow channels the length of attenuation of the secondary flow iS.;of the same order of magnitude as the channel width and as the length over which the source of the secondary flow, viz. the curvature of the ma1n flow, decreases from a rather high value 1n the bend to almost zero in the straight reaeh. Consequently, the advective influence of the main flow on the secondary flow can not be neglected in a mathematical model focused on the computation of the secondary flow.

The purpose of the present model, however, 1S the computation of the main velocity distribution and the shear stress at the bot tom. The mai~ velocity distribucion in fully developed curved flow is influenced considerably by the secondary flow (DE VRIEND, 1978), but experiments

on developing turbulent flow in curved shallow channels (see, for instanee,

DE VRIEND, 1976 and DE VRIEND AND KOCH, (977) show that this influence needs a rather long distance to establish. In addition, in a straight reach after a bend it takes a long distance for the main flow distribution to reach its fully developed straight channel shape (ROZOVSKII, 1961; see also Appendix 11). As a consequence of this retarded response of the ma1n flow, the local errors .in the secondary flow near the entrance and ehe exit of

a

bend caused by neglecting the advective influence of the main flow will not give rise to important errors in the main velocity distribution. Therefore the terms representing the advective

influence of the main flow~ón the secondary flow the computation of the main flow.

12

-For the determination of the bed shear.stress vector, and especially its direction, the secondary flow must be computed accurately, so then the advective influence of the main flow can not be neglected. The direction of the bed shear stress, however, will play no part ~n the actual computation process of the main

velocity. Hence this direction, and so the corresponding more accurate

secondary velocity components, can be computed after this ma~n velocity computation has been terminated. Further details of this accurate secondary flow computat10n are given in chapter 5.

Neglecting secondary flow inertia and the advective influence of the main flow on the secondary flow, the momentum equations in radial and vertical direction become

2

3 u OV ov av êv u

e;Re (~~ + v ~ + v ~ + W arm) - e;2Re m

r o</> m ai; s ai; s., r

(3.12)

(3.13)

So the simplifiecilongitudinal momentum equation

(3.8)

concerns

only the ma~n flow and the simplified vertical momentum equation

(3.13)

concerns only the secondary flow. The radial momentum equation

(3.

:

12),

however, concerns both the main and the secondary flow.

3.3.

Transformation to streamwise coordinates

Defining a,normalized, stream-oriented, curvilinear coordinate system

(n,

s,

ç),

in which s denotes the distance along the straamlines of the ma in flow normalized by Rand

n

the distance along the normal lines

c '

normalized by d*), the momentum equations

(3.8), (3.11)

and

(3.12)

and

IV

the equations of continuity

(3.3)

and

(3.6)

can be transformed. If u

m

are the horizontal velocity components in s- and n-direction, ~s identically equal to zero by definition.

IV IV and v + v m s • IV respect~vely, v m

~) s increases ~n the main flow direction, n from the left bank on

I'

.

,

- 13

-Without g1v1ng further details of the transformation, the transformed differential equations read

'\0 '\0 au u m m

----

_{as r}

n

o

(3.14) '\0 av aw s E '\0 + __s

=

0

---

- v

an

_rs s ar;; '\0 au a~ au

ap

_IJ2 '\0 m '\0 _{m E} '\0'\0 _m

e:Re(u

-a-

+ v v u + w

_a-ç)

₌

-+ U

m S s

an

r s m s

as

2 m

s

'\02 '\0 2 um

-~

2IJ2 2

aw

m E Re- + _{E 2 v} + E r

an

s

as

s 0 - ~ + _E2IJ2_{2 w} ar;; s 2

a

2

a

a~

whieh IJ2,

a

E and '\0 m E '\0

1n

= --

_{+ -2}

---

w

=

- -- +

-

u and 2, ar;;2

_an

r

an

m

an

r

_

m s s (3.15) (3.16) (3. 17) (3. 18)

where I/r and I/r denote the eurvature, normalized by I/R ,

s

n

e

of the atream'li.nes and the normal lines, respeetively*).

The transformation leads to a eonsiderable simplifieation of the "radial" momentum equation (3.17), wlieh now elearly shows the secondary flow to be eaused by the eurvat~re of the main flow and, to a mueh lower extent, by the longitudinal variation of the

vortieity of the máin flow.

*)

The eurvature of a eoordinate line is taken positive when the outward normal of this line is direeted opposite to the positive direetion of the other horizontal coordinate line.

·

,

• J 14

-4. Main flow oomputation

The mathematical system to he solved in order to compute the main flow consists of the two equations of continuity

(3.3)

and

(3.6)

and the momentum equations

(3.8), (3.12)

and

(3.13),

comhined with the integral condition of continuity

(2.21),

the houndary conditions

(2.22)

and

(2.23)

and a numher of inflow and outflow conditions to he discussed later. In spite of the simplifications, this is a fairly complex system of nonlinear partial differential equations requiring a numerical

solution procedure.

In the next sections the most importa~t elements of the iterative procedure used to solve the main flow are discussed separately. An overall review of the'procedure is given in section 4.8.

4.1. Vertical distrihution of the ma1n velocity

M~~ing use of the similarity approximation

(3.7),

the tangential momentum equation

(3.8)

ean he rewritten to

- -

-f

2 (u au - au e)

f

(au e -) a

F-eRe - -- + v -- + - uv + ERe v + - u + ERe ~ w u

r aep a~ r s at; r ar; s

_ _!_ ~

+,f(::~2

+ ~ dU) + ~

a

2

f

r aep ö~ r al; ar;2

(4.1)

If the main flow inertia term is made linear a.n

f

hy setting

(4.2)

tin which ] is a known estimate of

f

(for instance the distrihution found in the foregoing iteration step), this is a linear second-order ordinary differential equation for

f

as a function of r; that can he solved if estimates of u, v, v and wand of the tangential pressure,

s s

gradient are known .

In effect the tangential pressure gradient needs not to he known if the pressure 1S assumed to he hydrostatic.in accordance with one of the conclusions drawn from fully developed curved flow computations

(DE VRIEND,

1978).

In that .case the pressure gradient term in

(4.1)

does not depend on r;and as the equation was made linear in

f

hy (4.2),

, J _{- 15}

-it ean be rewritten as an equation in!' =

f/(- ~ ~):

_ "I2f' _ :'lf"

a

2- - -

-u _0_ - e:Rew u :!.L + {~ + ~ ~u :...e:Re~ (~ at:+

v ~

+ ~

üv)

+

aç2 s aç .at:2 r at: r a<j> at: r

au

E-~ ERev (-- + - u)}f'

s at: r -}

l'he quantity

f'

ean be solved from this equation and the

boundary eonditions

af'

f '

_.

I

_{ç=-} -}

-

O·

_'aç

I

_ç=o--0

, after whieh

f

is derived from the eondition

f

= 1 wh i.chfollows

direetly from the definition (3.7).

Henee

r=rtr

,sO that

f

is known without introdueing the pressure gradient as a

known quantity.

4.2. Depth-averaged maln veloeity field: differential equations

Averaging the tangential and radial momentum equations (3.8) and

(3.12) over the depth of flow yields

l ~ ~ aü - aü e:-- au e:- ~

-ERe

f

(r

d<ji+ v

af

+ r uv) + cRe vsf (~ + r u) + e:Rew s aÇ u

•

₃ -2 - ,,- - "Iv 3 ~ "Iv af 2

2"

u-2

R

_{f (}

U oV o ) R (

f

o -) R

f

E e r~+v"ä[ +e: e Vs "ä[+wsaçv -e: e -;-=

2 2- • - _

a

f

l

êv 3

a-_ ~ + E {( d v + ~ av) _ v __ _ _s

I

}-

2 ~ ~

at: at:2 r at: aÇ ç=-} aç ç=-} r2 <l<j>

(4.3)

(4.4)

(4.5)

(4.6)

,J

16

-The quantities u, v and p can be solved from these equations

and the depth-averaged equation of continuity (3.5) if estimates

of

f,

v and ware known.

s s

This will be done using the stream function/vorticity concept.

Eliminating the pressure from equations (4.6) and (4.7) yields

an equation that ean be written as a transport equation for the

vorticity of the depth-averaged flow field, which can be defined by

2 -

-e:

av

au

e:-,

'

w=---u

r

a~

at;

r (4.8)

If terms being an order O(e:2) smaller than the leading terms of the

same type are neglected, the vorticity transport equation reads

- -

-

-e:Ref2 (~ aw + ~ aw) + e:Rev f (aw + ~ ;;;) +

r

a ~

at;

s

at;

r ,af} -+w - w+ s al;

- a

- e:Reu ~

---

e aw r

at;

(4.9)

Correspondingly, til is approximated by

-au

e:-w ::- -

_at;

- - u

r (4.10)

The depth-averaged equation of eontinuity ~s satisfied if the stream

funetion ~ is defined by act> u = -"

at;'

1 alb (4.11) v

=

r a<j>

Then the following relation between <l> and UI ean b> derived from

definitions (4.8) and (4.11):

- til (4.12)

The depth-averaged main·velocity field ean be determined by solving

..

17

-set of boundary conditions.

It should be noted that it would be co.nsistent to neglect

the term with the second derivative with respect to ~ in equation (4.12), this term being an order O(E2) smaller than the leading term in this equation. If this term would be neglected, however, the depth-averaged flow would become purely "parabolic" (cf. PATANKAR AND SPALDING, 1972 and also McGUliRK, 1978), i.e. no

upstream influencing would be incorporated in the model. When the

term is maintained, however, some of the "elliptic" features of the

flow are accounted for*). This approaéh is somewhat similar to a method

used by PRATAP AND SPALDING (1976) (see also McGUIRK, 1978) ~n a

mathematical model of "par tialIy parabolic" f Low, in which longitudinal

diffusion is neglected and upstream influencing is incorporated through

an elliptic equation for the pressure.

4.3. Depth-averaged ma~n velocity field: boundary conditions

Regarding equation (4.12),boundary conditions for ~ must be given

at the sidewalls and at the inflow and outflow boundaries.

At

the

-sidewalls u ~nd v must be equal to zero, 50 that

0;

o

at ~ ±

B/2d

(4.13)

As equation (4.12) is second-order in ~, however, only two out of

these four conditions can be imposed on ~. The other two oonditions

-must be accounted for in the boundary conditions for "4>. The most

obvious possibility is to take ~ constanc along the sidewalls, thus

satisfying the condition of impermeability v

=

O. As only derivatives

of ~ are of interest, one of these constant values may be chosen arbitrarily. Therefore ~ is set equal to zero at the left bank:

4>1~=-B/2d = 0

(4.14)

.

*) Part of the upstream influencing drops out when neglecting

18

-Substituting definition (4.11) into the integral condition of continuity (2.21) yields

~1~=B/2d - ~1~=-B/2d

B

cl

whence ~1~=B/2d (4.15)

At the inflow boundary the tangential velocity distribution will mostly be given. Through definition (4.11) this can be translated

into acondition for ~,reading'

1;

J

-B/2d

ui.

_~nflow d~ (4.16)

From a physical point of v~ew it is not quite obvious to prescribe the velocity distribution at the outflow boundary; it would be preferabie to have a free boundary there, influencing the flow as little as possible. Therefore at the outflow boundary the condition

(4.17)

is imposed

The vorticity transpozt equation (4.9) can be solved when estirnates of ~, ~, v , wand

f

are known and one inflow condition and two

s s

lateral boundary conditions for .w are given. The inflow condition ean be derived from the approxi~ated definition (4.10) and the given velocity distribution at the inflow boundary. The lateral boundary conditions, however, are more problematic, since at the sidewalls no conditions for ;, but only for ~ and ~, and s.ofor the first dèrivates of

~,

are given. One of the possibilities to cope with this problem is

derive

-

sidewalls

to

w

at the from (4.12) , which reduces there to

-

é)2~

=

±

B/2d

w

-

--

for _~ (4.18)

é)~2

Various finite-difference representation of this equation were applied successfully to laminar.flow problems (ROACHE, 1972). A disadvancage

19

of this approach is that the boundary conditions for

w

vary during the iterative solution process'of the main flow equations

and may give rise to instabilities.

A more direct approach, which is especially fit for the present

channe L flow problem with its predominant tangential")velo:iity

component, is based on the approximate definition of the vorticity

(4.10). The following expression for ~ ean be derived ftom this

equation and the boundary condition for u at the left bank:

u

=

E;

J ril) dE; r -B/2d

(4.19)

Then the boundary condition for u at the right bank leads LO the

integral condition for

w

"B/2d

J rwds=O

-B/2d

(4.20)

Another integral condition for w 1S derived from (4.19) and the

integral cond ition of .corrt i.nui ty (2.21). It reads

B/2d di;

J

--B/2d r i;

J

-B/2d rwdt.: B d (4.21)

The two integral conditions (4.20) and (4.21) will be used Lnsteac of

conditions for ~ at the sidewalls when solving the vorticity transport

equation (4.9). Further detail~ on the solution of this equation are

given in Appendix IV.

4.4. Secondary flow: stream function equation

The secondary velocity components v and w must be solved from the

s s equation of continuity

av

s; c -- + - v ai;

r

s

aw

+ __s=o al; (4.22)

arid the transverse momedtum equations (3.12) and (3.13). In this

- 20

-Regarding the expressions g1v1ng the main flow curvatures as a

funetion of the depth-averaged veloeity eomponents (see Appendix 111)

l' 8 (u-2

I

{~(E ~

av +

EY

av

2-2)3/2

r

a~

a~

+ E V 2 ~) r - u au - au E--EV (- - + v + - uv)} r

a~

a~

r

=

(4.23)

=

- -2 I {cv-(c_u av + ~v- av _ u - u au - au E--c- <- <- - ) + u(- - + v - + - uv)}

(~2

+ E2y2)3/2

r a~

a~

r r a~ a~ r l'

n

(4.24)

2

and neglecting terms that are an order O(E ) smaller than the

leading terms of the same type, equation

(3.12)

ean be rewritten to

uv +e:..-) l'

n

an c2f

_I

_aw

+

2-

a

2

f

2 2

- .zs: + c- E V -- + E 'iJl Vs

a~

r

a~

aç2

(4.25)

So the radi.aI momentum equation 1S quite simil.or to the momentum

equation normal to the streamlines

(3.18),

but it contains additional

terms with

y

and 1/1' •

n

In order to reduce the number:of depende~t variables during the actual

c'o~putationaf the secondary flow, the same procedur e as

il1

së c.ti.qn

4.2

is used, i.e. the pressure is eliminated from equations

(4.25)

and

(3.13)

and the velocity components are replaeed by the secondary flow

vortieity

w

s' defined by

av s

aÇ

(4.26)

and the stream funetion of the secondary flow ~I, defined by

v

=

s I

a~1

r ao ;

w s

=

.L

~I r

a~

(4.27)

.

) - 21

-~quations (4.25) and (3.13) ean then be replaeed by the relation between Ijl' and

w

s

rlil

s (4.28)

and the vortieity transport equation

2 2

a

W 010 0 [Ü

ij_

1 ",-S e: ss" "uI --+_._+--= --+v o~2 r

a

~

a

ç2

a

(

r

at

2 -2 Re

2.L (~

aÇ

r 8 + e:uv) + r

n

o~

~2~

av

-

ow

a

f

~ e:Re(-~ v ~ + w ~ ~ -

f

s u + . s ~)

aç

s r s '"

2

ä"Ç

r

ä"Ç

äÇ

n

"ç

n

(4.29)

Assuming the secondary velocity eomponents ~n the advection terms of (4.29) ean be estimated by their values ~n the foregoing iteration step, equations (4.29) and (4.28) are linear differential equations for'w and ljI', whi.r.h eould be solved in suecession if explicit

s

boundary c.ond iti ons for 10 and Ijl' were given. The boundary conditions

. s

for 10 , however, must be formulated in terms 01 ljI' (ROACHE, 1972;

s

see also section 4.2), so that an iterative solution of (4.28) and (4.29) is necessary.

Another possi~ility is to combine equations (4.28) and (4.29) to one fourth-order equation for ljI', readi~g

~f'2 - 2 0f'

ov

s o2f

aw

sof

Re .::.J_ (~ + e:uv) - e:Re {- u (v ~ +

f

)

+ ~(\-l - + - -)}

oç

r

8

rn

rn

s

a

Ç

ä"Ç

s aç2

a~

aÇ

(4.30)

The stream function ljI' can be solved from this equation with the boundary eonditions

22

-•

•

=

O·

_{~ ac .}

~'I

_1;=-1

·

= 0 (4.31a)

=

0 (4.31b)

,I.'

I

+ 0 êljJ'

I

"'+ . = 0

'I' E;~ -B/2d = .. ;

af:

E;= -B/2d _-;(4.31c)

Thus the secondary flow problem has been reduced to the solution of one linear fourth-order differential equation with .boundar-yconditions

bei~g formulated expLi.citLy,.Although the solution of .t.his equati.on·may.be les

laborious than the simultane.ous solution of the equation of continuity (4.22) and the transverse momentum equations (4.25) and (3.13) or the iterative solution of equations (4.28) and (4.29), fully developed curved flow computations showed that the total amount of work to be done when using equation (4.30) is not less than when using (4.28) and

(4.29), the overall iteration procedure converging faster in the latter case (CRENG , LIN A:JD OU, 1976). Nevertheless, the stream function equation wjll be used, this equation working better whenfuunher

simplifications are introduced, as will be shown in the next sections.

4.5. Further simplification of the stream function equation

In the main flow equations the secondary velocity components figure only in the secondary flow advection terms. In fully developed curved flow these terms have a most important effect on the transverse distribution of the main velocity (DE VRIEND, 1978). In order to attain this effect, however, the secondary flow advection,must act upor1"the ma in flow over a rather long traj ect. For turbulent flow this was readily shown by comparing experimental data with computational results from a model in which secondary flow advection was neglected (DE VRIEND, 1976 and 1977; DE VRIEND AND KOCR, 1977): the differences between the measured and the computed mean velocity distributions gradually increased when moving along the bend. Therefore, a good description of the secondary

• J

23

-flow in a bend is important for the computation of the main flow, but a local, not too large error in the secondary flow at the

transition between two channel sections of different curvature will have no .important consequences, the more so as round these

transitions the secondary flow advection terms in equations

(4.3)

and·(4.9) are in effect an order O(e:) smaller than the main flow inertia

'K'

terms

J.

This implies that terms being only important close to these transitions, such as those with ~ and I/r , can be neglected with

n

respect to terms being important throughout the bend, such as those

wi.t h I/rs' In addition, the advective influence of the secondary flow

on the main flow is only important if the Dean number Rele: is not small (~E VRIEND, 1978), i.e. if the Reynoldsnumber ~s considerably larger than 1. Consequently, the inertia terms will be far the most important ones out of the source terms in the stream function equation

(4.30).

Regarding these arguments, the secondary flow equation ~n the ma~n flow computation system is truncated to

'df2 ~2 -Re _._-'dç r

e

(4.32)

or, if 1jJ i.sdefined by 1jJ' Re1jJ

= _

'df2 ~2 'd'ç r s

(4.33)

4.6. Vertical distribution of the stream function of the secondary flow As became evident from the fully developed curved flow computations

(DEVRIEND , 1978), a similarity approximation of the stream function of the secondary flow yields satisfactory results. Assuming such a similarity approximation to hold in developing flow, as well, the

*)

Further on in the bend ihis is no longer true, since the more the

fLow approaches its fully deve Loped stage, the less important these roain flow inertia terms become.

·

) 24

-following definition is made:

~ g(ç; s) (4.34)

in whieh g ~s assurned to depend only weakly on

cp.

Then equation (4.33) ean be rewritten as

(4.35)

This equation ean be eonsidered as a linear fourth-order differential

-equation forg if estirnates of Ijl and the souree term are known.

As g will be deterrnined in the ehannel axis, where the vertiealvelocity

components as well as horizontal diffasion will be negligible

(DE VRIEND,

1978), equation (4.35) ean be redueed to

(4.36)

The boundary conditions needed to solve g from this equation are

U; ~I

aç'r.;=-I

O'

,

gl

ç=O -,-0'

a2g1

aç2 ç=O =0

(4.37)

Since by .definition g = I, the solution of the system (4.36) and

(4.37) ean be determined as. fo lows:

ç ç Ç2 31 0 ç

gI = f dl;

j

dç f

f

dç - (I+ - ç - - ç3) f dç

J

dç

_} + l +] 2 2 _} -I<

(4.38)

·

,

₂₅

-When making the same simplifying assumptions in equations

(4.28)

and

(4

.

29),

they reduee to

and _~ ~2

dr,;

r

e

(4.40)

with the boundary eonditions

g!

_r,;=-I-

-0'

_,

_g!r,;=o

and

W!

_s

_r,;=-

1= r

_~ d2g1

2

dr,;

r,;9-1

{I)!

=0

s

[=0

_~

(4.41)

The iterative solution of this system is likely to take mueh more time than evaluating

(4.38)

and

(4.39)

.

4~(7i. Depth-averaged stream funetion of the seeondary flow

Making use of the similarity hypothesis

(4

.

34)

and the boundary eonditions for g and

f

(see

(4.35)

and

(4

.

4),

respeetively), the depth-averaged stream funetion equation to be derived from equation

(4.33)

beeomes

4- 3- I

"

g

l

,,2;;:

-(_d

1/1 _

2

_E::

_d

1/1) 0

(2

0 'I'

2

_E::

_dlJ!)

r

d~4

r a~3 +

r

aç

r,;=0

d~2

-

r

d~

+ 3 3

1

-(~

_~

)1

3 3 r

ar,;

Ir,;=o

dr,;

,

r,;=-1

2

-f

!

r,;=0

-2

u r e

(4

.

42)

The relevant boundary eonditions are

± B/2d=0; d~1

=0

a~

~=

± B/2d

(4

.

43)

-1/1 can be solved from this equation if estimates of g and of the souree term in (4.42) are known.

Averagi~ equations

(4.28)

and

(4.29)

over the depth of flow yields

2-a ~

a~2

26

-• J

e:

aw

s

aw \

aw

I

~

2

____ + s s

=

_~I

+ negligible terms

r

a~

a~

~=O

a~

~=_I

ç=O

rs

(4.45)

When attempting to express the depth-averaged vertical diffusion term in (4.45) in terms of wand its derivatives with respect to ~,

s

this turns out to be impossible, as becomes evident from equation

(4.44) and

(hl, \ (lcu' \

~/

~=O -

a

"

~s ~=_I

r (~

aF,;

e: r

(4.46)

, which follows directly from equation (4.28). Consequently, equation (4.45) can not be written as a differential equation

in w , so that a depth-averaged equivalent of the system (4.28) s

and (4.29) does not exist.

4.8. Iterative solution procedure for the ma~n flow

Making use of the elements described in the foregoing sections, the following ~.terative solution procedure can be drawn up for the main

flow:

1. Estimate the vertical distribution functions

f

and g

,

for instance by taking their low R~ynolds number limits

f

3 (I - ~2) 24 '(Ç7 . 5 111;3- 51;;) (4.47) =

2'

; g

=

19 - 7Ç +

-and the depth-averaged quantities 1jJ and W.' for instance by

setting 1jJ

=

0 and taking the fully developed straight channel flow distribution of w.

2. Solve the main flow stream function ~ from equation (4.12) with the boundary conditions (4.14) through (4.17).

3. Determine the depth-averaged main velocity components u and v and the streamline aurvature 1/1' from the stream function, us~ng

s (4.11) and (4.23), respectively.

·

,

₂₇

-4. Solve the depth-averaged stream function of the secondary

flow ~ from èquation (4.42) with the boundary conditions (4.43) 5. Solve the vertical distribution function of the main velocity

f

from equations (4.3) and (4.5) in the channel axis, with the

bbundary conditions (4.4).

6. Determine the vertical distribution function of the stream function

of the econdary flow

g

by evaluating (4.38) and (4.39).

7. Solve the mean flow vorticity

w

from equation (4.9) with the

integral conditions (4.20) and (4.21) (see also Appendix IV).

8. Repeat the procedure from 2. on, until a termination criterion

is satisfied. As the main flow is concerned here, it is obvious

to define a criterion for the main velocity, for instance

max with \5 « 1 {4.48)

in which ~(n) denotes the value of u ~n the n-th iteration step*).

The solution of the differential equations and the evaluation of the

integrals and the partial derivatives is carried out in..each iteration

step, making use of second-order finite-difference methods on a

vertically uniform, but horizontally non-uniform grid with finer

meshes near the ~idewalls. Numerical details wi1:~.not be discussed

here, all procedures used being well-known.

*)

By definition the overall mean value of u equals I in each cross-seetion,

- 28

-.

,

5. Secondary flow and bed shear stress computation.

5.1.

Extensive computation of the secondary flow

Af ter the ma1n flow computation has been accomplished, the secondary flow must be computed more accurately than before 1n order to obtain a good prediction of the bed shear stress vector. Then neither the effects being only important near a transition between two channel sections of different curvature nor the

advective influence of the main flow can be neglected. Consequently, the momentum equations to be solved read (see also (4.25))

-2 - av av 3

e2Ref2 (!!:_ + eUv) + e3Ref (~ __ s + ~ "z:-s)+ e Re (- fv ~ + w ~ af)

r s::'n r acj> 0." S r

n

s Clz;;

an 2 1 a;:; 2- a21" 2 2

+..:..L+e

f--+e

v.:::._L_+eV v

aE;

r acj> az;;2 1 s (5. 1)

- aw aw

3 u s - s

e Re

f

(r ~

+ var-) = (5.2)

Using definition (4.27), the stream function equation to be derived from these equations becomes

r +

" 1 2, 2, 1 :l3+,

eRef (~_o + ~ ~){ _ (~ + LL)}

=

È.f._ _ aw + v s:s: +

r acj> af;' r af,;2 ar;2 .ar; r acj> a

s

3

a+.2 ~2 Re.::..L (

=-

+ . ar; 'r 'S 2 - av "f ~ av uv -

a

-+' 'u s U a: U s e-) - eRe {w v :::_,L_ + ~- - - v )- - - - f}

rn s ar;2 r acj> s rn ar; rnar; (5.3)

or,

if

the similarity hypothesis (4.34) is adopted

a 4,p-' 3- a 2 ,1 a 2,p'_{- 2}

_s,

_ai')

4

-g_

_{(-- 2 ~~) +}

_~-~-

+

.Lil

'.l!._' ₊

'

.

29

-~,2 -2 Re !!.I_ (~ + E uv) al; r r

s

n

(5.4)

In the ehannel axis, where the stream funetion equation ~s solved to yield the vertical distribution funetion

g

,

lateral diffusion

n

is negligible and the n-th radial derivative is an order O(E )

smaller than near the sidewalls. This implies that equation (5.Lf)

ean be simplified to

af

2 -2 uv Re _._ (~ +E -) al; r r

s

n

(5.5)

If g <,in_the adve :tion terms is estimated by t:he distribution

~n the foregoing iteration step and the quantity q ~s defined by

q - 2 -F aw - a--F L - + V :::..._.[_-r

a~

_{, al;}

"

2 + E uv) 1'"

n

+ R

f ~ (~

a~' ~ ~

i')

E e ,al; 2

a~

r r r

n

(5.6)

, equation

(5.5)

ean be written as

(5.7)

For the relevant boundary eonditions (see 4.37) the solution of this equation ean be determined as follows (cf. seetion 4.6):

30

-.

,

r; I; r; _{3 ) 3} 0 r; r; gl

₌

_J' dZ; J dZ; f qdr;

-

() +-r;--I; ) J dr; J dr; J qdZ;+ -) -) -1

2

2

-) -) -) +

* ()

+ r;)

2

0 J qdr; (5.8) -) (5.9)

The depth-averaged stream function equation to be derived from (5.4)

and the boundary conditions for

f

and

g

reads

e:

a~

I

(~I _~I )~

I

--)_{r oE;} + 3 3 -_r +

ar;

ç=O

ar;

ç=-)

.

=

(5.)0)

This equation can be solved with four lateral boundary conditions

(see 4.43) and one inflow condition, for instance

~II

_s=O

=

0

(5.)))

If g and ~I are solved alternately, equations (5.8) through (5.11)

form an iterative solution procedure for equation (5.4). As g

will not be influenced very strongly by the advection terms

{cf. Appendix I), the iteration is likely to converge rather fast.

Therefore this'iterative approach will be preferabie to the direct

so~ution of equation (5.5), which is much more laborious than

- 31

-• J

5#2. Bed shear stress

The bed shear stress components TR and Tep are given by

TR =

n

::~l

z=-d and

Normalizing these quantities by

T

=

nV ,

R

d

r

nV

and

\rcp

=

d 'cp

, the quantities 'r and

'cp

can be elaborar.ed to

~ e ~

afl

-

eRe

i

al; l;=-1 r

- afl

u-al; l;=-1

The magnitude 'b and the direction with respect to the channel

.

6xis

a._"[ eau then be üetermined_{__.} from

,

=

eb . and (5.12) (5.13) (5.14) (5.15) (5.16)

.

,

"

32

-6. Verification of the model

6.1. Verification of the simplifying assumptions

The most important simplifying assumptions underlying the model are

the applicability of the simplified equation of continuity for the mairi flow (see section 3.1),

. the similarity hypothesis for the main flow, especially regarding the effect of streamwise accelerations,

• the simplifications of the stream function equation for the secondary flow during the main flow computation (see

sections 3.2 and 4.5).

It will be attempted to verify these assumptions, either by estimating the magnitude of the neglected terms or by including these tevlUS in the model and comparing the results with those

from the original model.

This verification will be carried out making use of the com-putational r0.sults for the LFM-flume (see section 6.2), which has a rather long, ~harp berid giving rise to strong effects of

curvature, tnus providing the possibility of testing the model

rather severaly.

6.1.1. The equation of continuity for the ma~n flow

On the basis of the assumptions made in section 3.1, the equation of continuity was split up into two parts, one for the main flow and one for the secondary flow. In the main flow equation (3.2) the term with the vertical velocity component was neglected to yield equation (3.6), which was used in the mathematical model. According to equation (3.5), this simplification is allowable if

~ :2t.

- ~

«

f. a~

f

av

(6. 1)

r

a~ ,

v

a~

r

a~'

a~

In order to indicate the !pplicability of

(6.11,

figure 4 represents

f.

dU u

af

33

-• J

as computed from the results of the laminar flow computation for the LFM-flume. This figure shows that, although (6.1) is not_true in every cross-section, the range of variation of

f

au. b 5' 1 h f" u af

r

,

äf

1.Sa out t1.mes as arge as t e range 0 var1.at1.onof

r

a;p'

Hence it is concluded that the equation of continuity for the main flow (equation 3.5) may be simplified by neglecting the

terms repr~senting the streamwise variation of

f.

6.1.2. The similarity hypothesis for the main flow

As far as the influence of diffusion and secondary flow advection is concerned, the applicability of a similarity hypothesis for the main flow was extensively verified for fully developed curved

laminar flow (DE VRIEND, 1978). In developing curved flow, however , not only diffusion and secondary flow advection will affect the main velocity distribution, but also streamwise accelerations of

the main flow will do 50. In general, positive streamwise

accelerations will give rise to a flattening of the vertical distribution of the main velocity and negative accelerations make

the main velocity profile more oblique (DE VRIEND, 1976 and 1977). As in a cross-section the streamwise accelerations of the main flow

will change sig.l (the overall mean value of the normalized velocity

equals 1 in every cross-section), they will disturb the similarity of the main flow ifi their effect is strong enough. Therefore the influence of these streamwise accelerations was investigated for the LFM-flume, making use of the results from the computations mentioned in section 6.1).

In the mathematical model, equation

(4.3)

is solved in the channel axis to yield the vertical distribution function

f,

which is

assumed to hold for the whole cross-section. The effect of streamwise accelerations on the vertical distribution of the main velocity

outside the channel axis can be estimated by considering the solution of

-

af'

- e::Re(wu) ~ s c al_;

+ [(a2~ +E- a;:i) _ e::Re~(~a;:i+ v au + e::uv)

a~2 r a~ c c r

a~

a~ r c

- cke {v (_au+ c ~)}

Jf'

= -)

(6.2)

,

.

₃₄

-with

f'/f'

=

f

and the suffix c indicating that the value in the channel ax~s ~s concerned.

When assuming

f

to be weakly dependent on ~ (i.e. its radial derivatives are negligible) and applying (6.2) instead of (4.3) in the last iteration step of the computations, the vertical distribution of the main velocity (figure Sa) is influenced considerably ln the first part of the bend and, to a lower extent, near the bend exit. Nevertheless, the mean velocity

•

distribution (figure Sb) is hardly affected, in contrast with the bed shear stress (figure Sc), in which the effect of the streamwi.se accelerations locally amounts 30%.

As a consequence, the effect of the streamwise accelerations on the similarity of the main fLow may be neglected when

computing the mean velocity field, but it must be incorporated in the bed shear stress computation, which can be done by solving equation (6.2) in each vertical of each cross-section.

It should be noted that, as

f

figures in the souree term of the secondary fl~w equation (5.4), the vertical distribution function g should be aLl.owel to vary with ~, as well. The influence of

the str eamwi.se accelerations through

f

on g can be accounted for

by evaluating

(

j

.B)

and (5.9) in each verticalJ computing the

souree term q by q ( _!_ ~) ,f ,r

a~

,

c + v C -2 u Re(-.1" 8 uv + E -) 1"

n

c 2 u

a~'

f

+ ERe( -2 13<1> r (6.3)

in which

g

~s estimated by the distribution ~n the channel axis found from the foregoing iteration step.

6.1.3. The simplification of the secondary flow equation

During the computation of the main flow the stream function ~) of the secondary circulation is determined by evaluating equations

(4.38) and (4.39), which yield the vertical distribution fllnction

g,

and by solving equation (4.42), yielding the distriblltion of ~

35

-The most important simplifying assumptions incorporated in these equations are

• the streamwise inertia of the secondary flow is negligible and

• the main flow curvature is the only source of secondary flow.

These assumptions could be verified by comparing the results

of a compu~ation with the extensive secondary flow equations

given in chapter 5 and the results of .a computation with the

truncated equations described in chapter 4.

When attempting to do so for the LFM-flume, however, the inertia

terms in the equation for g (see equations 5.6 and 5.7) turn out

to be negligible, except at the transitions be tween straight

and curved channel sections, where strong local effects are found. These effects are so strong that they disturb the

convergence of the iteration procedure, even if the terms are

brought to the left hand side of equation (5.7) instead of being

treated as known source terms. Therefore, regarding the small

influence aw~y from the transitions, the inertia terms were

omitted from the cOluputation of g. Thereby the iteration became

convergent, agà5n, and the verification could be carried out as

suggested before.

As regards the ma in velocity fields (figure 6), the effect of the

simplifications is rather small (less than 10%) and the essential

features of the main velocity distribution, such as the outward

shift of the mean velocity maximum and tne flattening of the

vertical distribution of the main flow, are hardly affected.

When considering the bed shear stress, however, these simplifications

appear to have a considerably stronger effect, on the magnitude of

this stress (up to 20%, see figure 7a), but especially on its

direction (locally up to 100, compared with an overall mean angle

of about 200; see figures 7b and 7c). As the bed shear stress factor

of the ma in flow (figure 7d) and the mean velocity field are hardly

affected, these differences must be attributed to differences in the

·

)

₃₆

-secondary flow.to be strong, indeed, not only in the transition zones between the bend and the straight reaches, but also halfway the bend. This strong effect halfway the bend seems surprising, S1nce the effect of the simplifications on the

mean velocity is rather small there. It should be noted, however, that the leading source term in the secondary flow equation is proportional to ~2, so that differences in u can give rise to considerably larger differenccs in ~. This is readily illustrated by the results of the extensive secondary flow computation on the basis of the main velocites resulting from the sirnplified model

(see figure 8): outside the transition zones between the bend and the straight reaches these results hardly differ from those obtained from the simplified model. Obviously, it is the interaction between

the secondary flow and the main flow that gives rise to the strong influence of the simplifications in the secondary flow equation halfway the bend.

As the secondary flow plays an important part in the bed shear stress, 1n its direction, but also in its magnitude (see figures 7a and 7d), the bed she1.r stress computation procedure proposed in chapter 5, making use of the wain velocites resulting from the simplified model, will not Lc app li.cable , either. This becomes evident from figures

7a and 7d, showing the magnitude of the bed shear stress resulting

from this procedure to be predicted hardly better than if the siwplified secondary flow equation were applied or even the component due to the secondary flow were neglected.

In summary, the foregoing confirms the presumption made in chapters 4 and 5, stating that the simplified secondary flow equation can be applied when computing the main velocity, but not when computing the bed shear stress. In order to reliably predict the bed shear stress, however, the extensive secondary flow equation should be applied not on1y when computing the bed shear stress, but a1so during the main flow computation.

6.2. Qualitative comparison with turbulent flowexperiments

37 -• j

is to compare them wi rh those from other models or from experiments. As no computational or experimental data on laminar flow in curved shallow channels are available 1n the literature and carrying out extensive laminar flowexperiments would &01 beyond the scope of the present investigations (aiming

at the mathematical prediction of turbulent flow), this way of

verification is not open here. Therefore only qualitative

comparisons with experimental data on turbulent flow will be made.

The data to be used for this qualitative verification of the

mathematical model were drawn from two series of experiments

on turbulent flow in curved channels, viz •

• experiments carried out at the Laboratory of Fluid Mechanics

of the Delft University of Technology in a 1.70 m wide flume

consisting of a 1800curved section with a radius of curvature

of 4.25 mand two straight inflow and outfLow sections of about

6 m length (see also DE VRIEND, 1976 and 1977),

• experiments carried out at the "De Voorst"-branch of the

Delft Hydraulics Laboratory in a 6.00 m wide flume consisting

of a 32 m long straight inflow section and a curved section

o

of about 90 with a radius of curvature of 50.00 m (DE VRIEND AND

KOCH, 1977).

These tW()fLumes will be referred to as the LFM-flume and the

DFL-flume, respectively.

The flow conditions in each of these flumes during the experiments

chosen for the verification of the present model are summarized

in the following table, i~ which DeT denotes the turbulent Dean

number

13C/s/lg

(DE VRIEND, 1978). The laminar Dean number

R B d Q C DeT c (m) (m) (m) 3 (m~Is) flume (m Is)

========-1=================== ======== ========== ======= =========

LFM 4.25 1.70 0.17 0.19 30 25

.

DHL 50.00 6.00 0.25 0.61 50 15